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UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations Title Aspects of Emergent Geometry, Strings, and Branes in Gauge / Gravity Duality Permalink https://escholarship.org/uc/item/8qh706tk Author Dzienkowski, Eric Michael Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California University of California Santa Barbara Aspects of Emergent Geometry, Strings, and Branes in Gauge / Gravity Duality A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Eric Michael Dzienkowski Committee in charge: Professor David Berenstein, Chair Professor Joe Polchinski Professor David Stuart September 2015 The Dissertation of Eric Michael Dzienkowski is approved. Professor Joe Polchinski Professor David Stuart Professor David Berenstein, Committee Chair July 2015 Aspects of Emergent Geometry, Strings, and Branes in Gauge / Gravity Duality Copyright c 2015 by Eric Michael Dzienkowski iii To my family, who endured my absense for the better part of nine long years while I attempted to understand the universe. iv Acknowledgements There are many people and entities deserving thanks for helping me complete my dissertation. To my advisor, David Berenstein, for the guidance, advice, and support over the years. With any luck, I have absorbed some of your unique insight and intuition to solving problems, some which I hope to apply to my future as a physicist or otherwise. A special thanks to my collaborators Curtis Asplund and Robin Lashof-Regas. Curtis, it has been and will continue to be a pleasure working with you. Ad- ditional thanks for various comments and discussions along the way to Yuhma Asano, Thomas Banks, Frederik Denef, Jim Hartle, Sean Hartnoll, Matthew Hastings, Gary Horowitz, Christian Maes, Juan Maldacena, John Mangual, Don Marolf, Greg Moore, Niels Obers, Joe Polchisnki, Jorge Santos, Edward Shuryak, Christoph Sieg, Eva Silverstein, Mark Srednicki, and Matthias Staudacher. I was very fortunate to have the opportunity to interact with many talented physicists. As such, I would like to thank the members of the Gravity / High Energy Theory group at UCSB and the KITP for their contribution to my edu- cation. A particular thanks to Joseph Polchinski and David Stuart for serving on my committee. I would like to thank my friends and fellow graduate students, especially those who reside with me on the sixth floor of Broida, and in particular, Sebastian Fis- chetti, Keith Fratus, and Eric Mintun. Thank you for the many useful discussions and insights, and for sharing many working nights. To my family, for their tremendous amount of support and understanding from v 2700 miles away while I accomplished this goal. Work of E.D. supported by the Department of Energy Office of Science Grad- uate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under contract no. DE-AC05-06OR23100. Work also supported in part by DOE grant DE-FG02-91ER40618, and grant DE-SC0011702. Work also supported in part by the National Science Foundation grant no. Phy05-51164, and grant no. PHYS- 1066293. Work additionally supported by the FWO - Vlaanderen, Project No. G.0651.11 and the Odysseus program, by the Federal Office for Scientific, Techni- cal and Cultural Affairs through the Interuniversity Attraction Poles Programme Belgian Science Policy P7/37, the European Science Foundation Holograv Net- work and by a grant from the John Templeton Foundation. The opinions ex- pressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. The research leading to some of these results has received funding from the European Research Council under the European Communitys Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. [247252]. vi Curriculum Vitæ Eric Michael Dzienkowski Education 2015 Ph.D. in Physics (Expected), University of California, Santa Barbara. 2010 M.S. in Physics, Rensselaer Polytechnic Institute. 2010 B.S. in Physics and Mathematics, Rensselaer Polytechnic In- stitute. Publications [1] E. Dzienkowski, Excited States of Open Strings from N = 4 SYM, arXiv:1507.01595 [2] D. Berenstein, E. Dzienkowski, and R. Lashof-Regas, Spinning the Fuzzy Sphere, arXiv:1506.0172. [3] D. Berenstein and E. Dzienkowski, Giant gravitons and the emergence of geometric limits in beta-deformations of N = 4 SYM, JHEP 1501 (2015) 126, [arXiv:1408.3620]. [4] D. Berenstein and E. Dzienkowski, Numerical Evidence for Firewalls, arXiv:1311.1168. [5] D. Berenstein and E. Dzienkowski, Open spin chains for giant gravitons and relativity, JHEP 1308 (2013) 047, [arXiv:1305.2394]. [6] C. T. Asplund, D. Berenstein, and E. Dzienkowski, Large N classical dynam- ics of holographic matrix models, Phys. Rev. D87 (2013), no. 8 084044, [arXiv:1211.3425]. [7] D. Berenstein and E. Dzienkowski, Matrix embeddings on flat R3 and the geometry of membranes, Phys. Rev. D86, 086001 (2012) [arXiv:1204.2788]. vii Abstract Aspects of Emergent Geometry, Strings, and Branes in Gauge / Gravity Duality by Eric Michael Dzienkowski We explore the emergence of locality and geometry in string theories from the perspective of gauge theories using gauge / gravity duality. First, we explicitly construct open strings stretched between giant gravitons in N = 4 SYM. We find that these strings satisfy a relativistic dispersion relation up to three-loop order and conjecture that this should hold to all loop orders. We find the explicit dual solution to the string sigma model and find exact agreement with the geometric nature of the SYM operator and dispersion relation. Using these open strings as probes, we explore the local field theory on the worldvolume of the giant gravitons. Second, we use classical configurations in holographic matrix models to un- derstand the emergence of geometry from matrix coordinates. We construct an effective Hamiltonian for a probe brane that observes the geometry in a back- ground matrix configuration from which we can construct membranes embedded in three dimensional space. Adding angular momentum to these configurations we are able to observe continuous topology changes. We also study the classical evolution of holographic matrix models to generate a microcanonical ensemble of configurations and study their thermal and chaotic behavior. We argue that these thermal configurations are dual to black holes. viii Contents Curriculum Vitae vii Abstract viii 1 Introduction 1 1.1 Permissions and Attributes . .8 Part I Open Strings in AdS / CFT 10 2 Introduction 11 3 Anomalous Dimensions of Open Strings 17 3.1 The su(2) Sector . 17 3.2 Giant Gravitons and Open Strings . 20 3.3 Strings as Cuntz Oscillator Chains . 25 3.4 The Cuntz Hamiltonian . 30 3.5 Discussion . 52 4 Exploring the Ground State 56 4.1 Finding the Ground State . 56 4.2 Geometry from Giant Gravitons and Strings . 59 4.3 Correcting the Ground State . 65 4.4 A Relativistic Dispersion Relation . 80 4.5 The Open String Dual . 92 4.6 Discussion . 98 5 Emergence of Geometric Limits in N = 4 SYM 102 5.1 Marginal Deformations of N = 4 SYM . 102 ix 5.2 The β-Deformed Cuntz Chain . 109 5.3 Geometric Limit Interpretation . 113 5.4 Discussion . 121 Part II Geometry from Matrix Models 123 6 Introduction 124 7 The Geometry of Membranes 131 7.1 Orbifolding The BFSS and BMN Matrix Models . 131 7.2 The Index: Adding a D0-brane Probe . 137 7.3 Fuzzy Spheres and Emergent Surfaces . 147 7.4 A Linking Number . 154 7.5 Discussion . 161 8 Classical Dynamics of Holographic Matrix Models 165 8.1 Observables and Symmetry . 165 8.2 Numerical Implementation . 172 8.3 Thermalization . 177 8.4 Power Spectra and Classical Chaos . 188 8.5 Factorization . 200 8.6 Discussion . 206 9 Black Holes From Matrix Models 208 9.1 Gravity From Matrix Models . 209 9.2 Aspects of Matrix Black Holes . 214 9.3 Exploring the Gapless Region . 218 9.4 Locating the Horizon . 226 9.5 Discussion . 229 10 Adding Angular Momentum 232 10.1 The Hamiltonian and the Ansatz . 233 10.2 Symmetry Considerations . 240 10.3 The Solutions as a Set of Critical Points . 245 10.4 The case of 2 × 2 matrices . 251 10.5 The case of 3 × 3 matrices . 255 10.6 Other Examples . 264 10.7 Large N ................................ 266 10.8 Topology Change . 274 x 10.9 Discussion . 278 11 Conclusions 282 A Conventions and Relations for the Lie Algebra of U(N) 285 B Cuntz Algebra and Hamiltonians 288 B.1 Cuntz Oscillators . 288 B.2 Closed Cuntz Hamiltonians . 290 B.3 Open Cuntz Hamiltonians . 291 C Proof of First Order Ground State Correction 293 D BFSS and BMN model conventions 295 E Fermion Decomposition 298 F Fermionic Modes Between Displaced Fuzzy Spheres 300 F.1 Diagonal Fermionic Modes . 300 F.2 Off-diagonal modes . 303 G Measuring the Temperature 306 Bibliography 308 xi Chapter 1 Introduction Quantum mechanics has been a powerful framework for building physical models of the universe that describe phenomena which occur at length scales comparable to that of an atom. It has a tremendous range of applicability, from the electron- ics in our mobile devices to processes occurring at the center of our Sun. It has revolutionized the way we understand particles and measurements. We learned that particle and wave like behavior have a unified description called the wave function. The act of measuring once can change the outcome of future measure- ments. These notions challenge our classical understanding of the world and have truly affected the way we interpret reality. As we probed nature over the past century, we determined that many of the observed phenomena were due to the following four fundamental interactions: electromagnetic, weak, strong, and gravitational. The first three of these interac- tions have a very good quantum mechanical description called the Standard Model of particle physics.
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